Two matrices whose product is equal to the identity matrix

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I need to multiply two matrices one we call P type 1x4 another called Q type 4x1

I cannot find values that will give me an identity matrix as a result when I multiply PQ together.

Where zeros are not allowed.

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Take $P=(1\;1\;1\;1)$ and $Q=\frac14^{\mathrm t\!}P$.